Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×D8 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,8,12,10,15,23,27,11,16,24,28,9,25,19,14)(7,17,20,13,22,26,36,30,18,33,34,31,21,29,35,32);(1,2,3,4)(6,12,15,27,16,28,25,14)(7,18)(13,29,30,17,31,26,32,33)(20,34)(21,22)(35,36);(5,7,19,35,9,21,24,34,11,18,23,36,10,22,8,20)(6,13,12,26,15,30,27,33,16,31,28,29,25,32,14,17)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eg,d^2=gi,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286801, #72286915, #72286922, … |
| maximal normal subgroup of | #72286801, #72286915, #72286922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47928);